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Theorem mircgrextend 28691
Description: Link congruence over a pair of mirror points. cf tgcgrextend 28494. (Contributed by Thierry Arnoux, 4-Oct-2020.)
Hypotheses
Ref Expression
mirval.p 𝑃 = (Base‘𝐺)
mirval.d = (dist‘𝐺)
mirval.i 𝐼 = (Itv‘𝐺)
mirval.l 𝐿 = (LineG‘𝐺)
mirval.s 𝑆 = (pInvG‘𝐺)
mirval.g (𝜑𝐺 ∈ TarskiG)
mirtrcgr.e = (cgrG‘𝐺)
mirtrcgr.m 𝑀 = (𝑆𝐵)
mirtrcgr.n 𝑁 = (𝑆𝑌)
mirtrcgr.a (𝜑𝐴𝑃)
mirtrcgr.b (𝜑𝐵𝑃)
mirtrcgr.x (𝜑𝑋𝑃)
mirtrcgr.y (𝜑𝑌𝑃)
mircgrextend.1 (𝜑 → (𝐴 𝐵) = (𝑋 𝑌))
Assertion
Ref Expression
mircgrextend (𝜑 → (𝐴 (𝑀𝐴)) = (𝑋 (𝑁𝑋)))

Proof of Theorem mircgrextend
StepHypRef Expression
1 mirval.p . 2 𝑃 = (Base‘𝐺)
2 mirval.d . 2 = (dist‘𝐺)
3 mirval.i . 2 𝐼 = (Itv‘𝐺)
4 mirval.g . 2 (𝜑𝐺 ∈ TarskiG)
5 mirtrcgr.a . 2 (𝜑𝐴𝑃)
6 mirtrcgr.b . 2 (𝜑𝐵𝑃)
7 mirval.l . . 3 𝐿 = (LineG‘𝐺)
8 mirval.s . . 3 𝑆 = (pInvG‘𝐺)
9 mirtrcgr.m . . 3 𝑀 = (𝑆𝐵)
101, 2, 3, 7, 8, 4, 6, 9, 5mircl 28670 . 2 (𝜑 → (𝑀𝐴) ∈ 𝑃)
11 mirtrcgr.x . 2 (𝜑𝑋𝑃)
12 mirtrcgr.y . 2 (𝜑𝑌𝑃)
13 mirtrcgr.n . . 3 𝑁 = (𝑆𝑌)
141, 2, 3, 7, 8, 4, 12, 13, 11mircl 28670 . 2 (𝜑 → (𝑁𝑋) ∈ 𝑃)
151, 2, 3, 7, 8, 4, 6, 9, 5mirbtwn 28667 . . 3 (𝜑𝐵 ∈ ((𝑀𝐴)𝐼𝐴))
161, 2, 3, 4, 10, 6, 5, 15tgbtwncom 28497 . 2 (𝜑𝐵 ∈ (𝐴𝐼(𝑀𝐴)))
171, 2, 3, 7, 8, 4, 12, 13, 11mirbtwn 28667 . . 3 (𝜑𝑌 ∈ ((𝑁𝑋)𝐼𝑋))
181, 2, 3, 4, 14, 12, 11, 17tgbtwncom 28497 . 2 (𝜑𝑌 ∈ (𝑋𝐼(𝑁𝑋)))
19 mircgrextend.1 . 2 (𝜑 → (𝐴 𝐵) = (𝑋 𝑌))
201, 2, 3, 4, 5, 6, 11, 12, 19tgcgrcomlr 28489 . . 3 (𝜑 → (𝐵 𝐴) = (𝑌 𝑋))
211, 2, 3, 7, 8, 4, 6, 9, 5mircgr 28666 . . 3 (𝜑 → (𝐵 (𝑀𝐴)) = (𝐵 𝐴))
221, 2, 3, 7, 8, 4, 12, 13, 11mircgr 28666 . . 3 (𝜑 → (𝑌 (𝑁𝑋)) = (𝑌 𝑋))
2320, 21, 223eqtr4d 2786 . 2 (𝜑 → (𝐵 (𝑀𝐴)) = (𝑌 (𝑁𝑋)))
241, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23tgcgrextend 28494 1 (𝜑 → (𝐴 (𝑀𝐴)) = (𝑋 (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cfv 6560  (class class class)co 7432  Basecbs 17248  distcds 17307  TarskiGcstrkg 28436  Itvcitv 28442  LineGclng 28443  cgrGccgrg 28519  pInvGcmir 28661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-trkgc 28457  df-trkgb 28458  df-trkgcb 28459  df-trkg 28462  df-mir 28662
This theorem is referenced by:  mirtrcgr  28692
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